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人気のある微分積分 >

limit as x approaches infinity of (sqrt((x+1)^2+1))/(sqrt(x^2+1))

解

x \to \infty lim \frac{( x + 1 ) ^{2} + 1}{x ^{2} + 1}

解

1

解答ステップ

x \to \infty lim \frac{( x + 1 ) ^{2} + 1}{x ^{2} + 1}

簡素化

\frac{( x + 1 ) ^{2} + 1}{x ^{2} + 1} : \frac{x ^{2} + 2 x + 2}{x ^{2} + 1}

= x \to \infty lim (\frac{x ^{2} + 2 x + 2}{x ^{2} + 1})

x \to a lim [f (x)]^{b} = [x \to a lim f (x)]^{b}

不定形式の場合を除く

= x \to \infty lim (\frac{x ^{2} + 2 x + 2}{x ^{2} + 1})

最高の分母べき乗で割る:

\frac{1 + \frac{2}{x} + \frac{2}{x ^{2}}}{1 + \frac{1}{x ^{2}}}

= x \to \infty lim (\frac{1 + \frac{2}{x} + \frac{2}{x ^{2}}}{1 + \frac{1}{x ^{2}}})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

不定形式の場合を除く

= \frac{lim _{x \to \infty} ( 1 + \frac{2}{x} + \frac{2}{x ^{2}} )}{lim _{x \to \infty} ( 1 + \frac{1}{x ^{2}} )}

x \to \infty lim (1 + \frac{2}{x} + \frac{2}{x ^{2}}) = 1

x \to \infty lim (1 + \frac{1}{x ^{2}}) = 1

= \frac{1}{1}

簡素化

= 1

グラフ

人気の例

limit as x approaches 1 of (sqrt(10-x)-sqrt(8+x))/(x^3-1)

x \to 1 lim (\frac{10 - x - 8 + x}{x ^{3} - 1})

limit as x approaches-3+of-(sin(c-3x))/(e^{5x+15)-1}

x \to - 3 + lim (- \frac{sin ( c - 3 x )}{e ^{5 x + 15} - 1})

limit as x approaches 0+of (e^x-1-x)/(x^2)

x \to 0 + lim (\frac{e ^{x} - 1 - x}{x ^{2}})

limit as t approaches infinity of te^t

t \to \infty lim (t e^{t})

limit as x approaches 0 of 3/2

x \to 0 lim (\frac{3}{2})